concept 3d vector in category python

This is an excerpt from Manning's book Math for Programmers: 3D graphics, machine learning, and simulations with Python MEAP V11. Login to get full access to this book.

The most important part of this equation is the symbol that looks like an upside-down triangle, which represents the gradient operator in vector calculus. The gradient of the pressure function p(x, y, z) at a given spatial point (x, y, z) is the 3D vector q(x, y, z), indicating the direction of increasing pressure and the rate of increase in pressure at that point. The negative sign tells us that the 3D vector of flow rate is in the opposite direction. This equation states, in mathematical terms, that fluid flows from areas of high pressure to areas of low pressure.

In a 2D space, like a page of this book, we have a vertical and a horizontal direction. Adding a third dimension, we could also talk about points outside of the page or arrows perpendicular to the page. But even when programs simulate three dimensions, most computer displays are two-dimensional. Our mission in this chapter is to build the tools we need to take 3D objects measured by 3D vectors and convert them to 2D so our objects can show up on the screen.

Figure 3.18: A second application of the Pythagorean theorem gives us the length of the 3D vector.

This is an excerpt from Manning's book Natural Language Processing in Action: Understanding, analyzing, and generating text with Python. Login to get full access to this book.

What about 3D vector spaces? Positions and velocities in the 3D physical world you live in can be represented by x, y, and z coordinates in a 3D vector. Or the curvilinear space formed by all the latitude-longitude-altitude triplets describing locations near the surface of the Earth.

These six topic vectors (shown in Figure 4.1), one for each word, represent the meanings of your six words as 3D vectors.

Figure 4.1. 3D vectors for a thought experiment about six words about pets and NYC

Earlier, the vectors for each topic, with weights for each word, gave you 6-D vectors representing the linear combination of words in your three topics. In your thought experiment, you hand-crafted a three-topic model for a single natural language document! If you just count up occurrences of these six words and multiply them by your weights you get the 3D topic vector for any document. And 3D vectors are fun because they’re easy for humans to visualize. You can plot them and share insights about your corpus or a particular document in graphical form. 3D vectors (or any low-dimensional vector space) are great for machine learning classification problems, too. An algorithm can slice through the vector space with a plane (or hyperplane) to divide up the space into classes.

But don’t you think good old linear SVD and PCA do a pretty good job of preserving the “information” in the point cloud vector data? Doesn’t your 2D projection of the 3D horse provide a good view of the data? Wouldn’t a machine be able to learn something from the statistics of these 2D vectors computed from the 3D vectors of the surface of a horse (see figure 4.5)?